Abstract

This paper is devoted to the investigation of nonnegative solutions and the stability and asymptotic properties of the solutions of fractional differential dynamic linear time-varying systems involving delayed dynamics with delays. The dynamic systems are described based on -calculus and Caputo fractional derivatives on any order.

1. Introduction

There are abundant background results concerning the exact and approximate solutions of fractional differential equations [1–4], fractional derivatives involving products of polynomials [5, 6], fractional derivatives and fractional powers of operators [7–9], and boundary value problems concerning fractional calculus in a theoretical context and also concerning a wide range of applications like, for instance, control theory, robotics, signal processing, heat transfer, lossless transmission lines, and so forth [1–22]. In particular, some generalized operators of fractional integration have been recently applied to the product of generalized Bessel functions of the first class in [6] leading to multivariable generalized Lauricella-type functions. Furthermore, related generalized fractional integrals are also discussed in that paper. On the other hand, new unified integral formulas involving products of Srivastava-type polynomials and the -function as well as fractional integration for Appell’s functions are discussed in [23, 24].

This paper is concerned with the investigation of nonnegative solutions of fractional -differential dynamic systems with point delays and some related asymptotic properties formulated by the Caputo fractional derivative. See, for instance, [25–31] for formulations of functional dynamic systems under delays. Some of these papers are concerned with fundamental properties of positive dynamic systems or with the nonnegative solutions of dynamic systems in a fractional context. See, for instance, [32–34]. On the other hand, see [13–22, 32–39] for a background on quantum and fractional calculus and some related applications to dynamic systems.

The Caputo -difference scheme has been proposed in [40] and then the problems of initial values are investigated in [41, 42]. In particular, a Caputo-type -fractional initial value problem is solved in [41] with the solution being formulated in terms of a new -Mittag-Leffler function. On the other hand, the related investigation in [42] is focused on analytical aspects of -fractional calculus while the variational iteration method is extended “ad hoc” to -fractional calculus in order to solve the Caputo -fractional initial value problem. There is also a recent monograph [43] available on the fractional -difference methodology which is of potential interest for readers interested in quantum fractional calculus. Also, it has to be pointed out that an increasing research interest is being devoted to the use of fractional calculus in the analysis of mathematical models based on partial differential equations. In particular, the fractal heat conduction problem is solved by proposing a local fractional variation iteration method in [44]. On the other hand, the solutions of the Helmholtz equation involving local fractional derivative operators are investigated in [45] combined with series expansion and variational iteration methods.

It might be pointed out that positive dynamic models are an essential tool to describe some real world applications as, for instance, medical, biological, or epidemic models. It has to be pointed out that a major advantage of the use of -calculus is that it does not need the existence of limits or restrictive regularity conditions on the functions dealt with in order to establish the formulation. In particular, derivatives and higher-order derivatives of a wide class of functions exist almost everywhere under the -calculus framework [37]. In this context, the -calculus formalism on differ-integral systems is close to the classical one on difference systems with the additional advantage that the parameter running the samples can be chosen to be real so that it links the selection of the sampling points in a multiplicative fashion. Such sampling points are backward-in-time dependent on each time instant for which the -fractional solution is evaluated while asymptotically vanishing to zero as the number of used samples increases to infinity for each given time instant where the quantum fractional solution is being computed.

1.1. Notation

, , and are the sets of integer, real, and complex numbers, and are the positive integer and real numbers, and
The following notation is used to characterize different levels of positivity of matrices.

is the set of all real matrices of nonnegative entries. If then is used as a simpler notation for .

is the set of all nonzero real matrices of nonnegative entries (i.e., those with at least one of their entries being positive). If then is used as a simpler notation for .

is the set of all real matrices of positive entries. If then is used as a simpler notation for . The superscript denotes the transpose and and are, respectively, the th row and the jth column of the matrix .

A close notation to characterize the positivity of vectors is the following.

is the set of all real vectors of nonnegative components. If then is used as a simpler notation for .

is the set of all real nonzero vectors of nonnegative components (i.e., at least one component is positive). If then is used as a simpler notation for .

is the set of all real vectors of positive components. If then is used as a simpler notation for .

is a Metzler matrix if . is the set of Metzler matrices of order .

The maximum real eigenvalue, if any, of a real matrix , is denoted by .

, , and mean, respectively, , , and for being any real scalars, vectors, or matrices of compatible dimensions or orders.

The following fundamental result of [32] is concerned with the unique left-sided solution on of the differential fractional system (36).

Theorem 1. Consider the Caputo fractional differential system of order with (potentially repeated) delays and distinct delays:
on , for any with if and if ; , are distinct constant delays, , are the matrices of dynamics for each delay , is the control matrix and with initial condition of the state being given by -real vector functions , with , which are absolutely continuous except eventually in a set of zero measure of of bounded discontinuities with and is a bounded piecewise continuous control function. Then, the unique left-sided solution of (2) is given by
with if and if , where
for and for , where are the Mittag-Leffler functions.

If -calculus is used then the integral formulas of Theorem 1 are not necessary to calculate the fractional Caputo-type solutions of any order as shown in the subsequent example.

Example 2. Consider the differential equation , . The standard solution for any real and is . It is globally stable (resp., globally asymptotically stable) for (resp., for ) and nonnegative for any if .The use of -calculus for yields
then, the solution becomes
(a)If then it is positive if and but any -solution is unbounded if and if and if .(b)If then so that the -solution is constant, globally stable, and nonnegative if .(c)If then ; with the first inequality being obvious and the second one being proved by contradiction. Assume that the second inequality is false. Then, there is , so that there is , such that for any , the subsequent contradiction is got:

Since is bounded for for any given then . Therefore, for the -differential equation has a nonnegative -solution on for any and it is globally asymptotically stable. On the other hand, the Caputo fractional solution of real order of the associate fractional differential equation is
The formulation within the -calculus framework leads to
Note that the solution is positive for any order if and for and . Assume that, for , there is some such that and for and some since it follows from continuity arguments that if it is negative at a point it is also negative on some interval containing the point. Then, the following contradiction arises:
so that any Caputo fractional solution of any real order is nonnegative for and any given initial conditions for . Since any solution is nonnegative for any nonnegative initial conditions, then if , one gets for the case of interest for
and then global (resp., global asymptotic) stability holds for (resp., for ) while the solutions are nonnegative for any nonnegative initial conditions and any real . These stability properties are independent of the nonnegativity of the solutions since if then by continuity of the solutions in some interval so that
and is large enough, since is strictly increasing since it possesses a nonnegative integrand, so that there is such that and then ; , that is, the solution reaches the zero-equilibrium in finite time and global asymptotic stability is guaranteed. To evaluate the error between the -calculus solution and the standard one, denote them, respectively, as and and then define the extended vector . The substitution of both solutions yields
Note that for , the first eigenvalue of the matrix of dynamics tends to if and to zero if while the second one converges to zero as in both cases. Thus, the error between both solutions converges asymptotically to zero if . If then , .

2. Preliminaries on Fractional -Differential Systems

Fundamental definitions of -calculus are [35–38]
The -power function is
If then the -power function is
leading, in particular, to . Formula (16) is the -analog of the Pochhammer symbol (-shifted factorial) [36]. The -derivative of a function is defined by
[35, 36] and the -derivative of high -order of a function is defined by and

Lemma 3. The following properties hold.(i)The commutation property ; , holds for any real .(ii)The commutation property ;, holds for for any ,, such that , provided that exists for any .(iii) or any real , and also for if exists for any .(iv)
for all , holds for any real and also for if
exist for all .

Proof. First, note that for the identity always holds for , with , which ensures the existence of for and . Proceed now by complete induction and assume that, for some given , the commutation property below holds for the existing for any real :
Then, by using the commutation property of the operator on for any real
The commutation of the operator composition for any nonzero has been proven. Now, for any integer we can find integers (being nonunique for ) such that . Hence, and the result follows by taking an arbitrary and nonzero . Property (i) has been proved. Property (ii) can be proved in the same way for and any such that exists for . Properties (i)–(iii) have been proved.On the other hand, the -derivative operator (17) is a linear operator [37], so that
Property (i) and (23) yield for , and also for if exists for any , the following relationships:
Hence, Property (iii) follows.To prove Property (iv), define the time -delay operator on as so that , with . Now, for any nonzero real , assume that the property is true for and some given . Thus,
for from (19) (first identity) and the definition of the operator (second identity). Then, one gets from the definition of the -derivative in (17) the use of the first and second identities of (25) for , with , the identity , and, finally, the second identity in (25) for :
so that if (19) holds for any given real and for any given then it also holds for and such a nonzero real . Then, by complete induction, (19) is true for any nonnegative integer . If exists for any the result also applies for .

Assume that for some real interval . Then, the Riemann-Liouville left-sided fractional -derivative of order of the vector function in is point-wise defined as
where and ∖, where , is the -gamma function defined as ; ∖ which satisfies the following relations [35, 36]:
where for is a -real number. Remember that the usual gamma function is defined by ; ∖. Now, we can replace the standard simple and higher-order derivatives under the integral symbol by their -derivative versions to build the Caputo fractional -derivative by using the identities
in ; then the Caputo left-sided fractional -derivative of order of the vector function in is point-wise defined as
where if and if . Note that the existence of in (30) is not required, as it is required in the standard fractional calculus for the existence of Caputo derivatives since the existence of the standard and higher-order fractional -derivatives is ensured.

Example 4. Consider the differential dynamic system
where are distinct constant delays, are the matrices of dynamics for each delay , is the control matrix and with initial condition of the state being given the real vector function , which is absolutely continuous except eventually in a set of zero measure of of bounded discontinuities with , and is a bounded piecewise-continuous control function. Some results about the nonnegativity of the solutions of (31) by using standard, fractional and -calculus follow below.

Proposition 5. The solution of (31) is nonnegative on for any and if and only if, and .

Proof. The sufficiency part of the proof is direct since the unique left-sided mild solution is given by
It turns out that, under the given assumptions, and then the solution is nonnegative; since ; since . On the other hand,
It is easy to see that it is always possible to get by construction if any of the conditions , ; , , or fails for some and by taking some large component in either the initial condition function or the control function corresponding to a negative entry of the matrix whose positivity condition fails.

Proposition 6 (Theorem 4.1(iii) of [32]). Any solution (37) to any Caputo fractional differential system of fractional order is nonnegative independent of the delays; that is, for some , for any set of delays satisfying and any absolutely continuous functions of initial conditions , and any piecewise continuous control , if and only if for being sufficiently small. Furthermore, either if, in addition, or if is nilpotent or if , () and .

Proposition 7. Consider the -calculus version of (31) under similar initial conditions:
Assume that , , . Then,(i); if either or and for all (≠i).(ii); if with ; and with for only one for each .

Proof. The solution of (34) is
provided that exists. Since , , , for nonnegativity of the solutions the existence of is also required. Note that if and nonsingular then if and only if is monomial (or generalized permutation matrix, i.e., it has only a nonzero entry per row and then only a nonzero entry per column). Due to its structure, this condition can be fulfilled with being diagonal, that is, for in order that is monomial and . Then, if and only if for all either or . Hence, Property (i) follows directly. Property (ii) follows since, under the given constraints, , with , and Metzler, monomial (then nonsingular) so that . Hence, Property (ii) follows.

3. Fractional -Differential Dynamic Systems of Order with Internal Point Delays

The fractional Caputo -differential dynamic systems of order can become modified from (2). The functions , , and are now displayed to be used later in order to solve such a differential system. The combination of (2) with (24) with the replacement of with leads to
(a)If and then , and
(b)If then , and
Now consider, from (2) and (24), the linear time-varying differential functional left-sided Caputo fractional -differential system of order
with ; , being distinct constant delays; , , are the bounded matrix functions of dynamics for each delay , and is the control matrix function. The initial conditions are given by -real vector functions , with and with , and is a bounded piecewise continuous control function. The Jackson integral in the integral term in (39) becomes
for . By replacing the dummy argument under the two integral symbols in (40) by and , respectively, since (see [37]):
The series in (41) is convergent for all since the function satisfies in a right neighborhood of for some real constants and . So, the Jackson integral in (40) is convergent since in some right neighbourhood of for real constants and [37, 39]. Equations (40) and (39), together with (18), Lemma 3(see (19)), and (17), yield:
The following auxiliary results hold concerning the product and the series .

Lemma 9. Define . Then, , where(a) and , , so that , if ;(b) and , , if ;(c).

Proof. Note that(1);(2)if ;(3)if , with , then , , , with and . Assume this is false so that . Then, and, equivalently, , or if (a contradiction).

Lemma 10. Assume that is bounded in a right neighborhood of for some and some real constant . Define
Then, the following identities hold:
for any given if , where , , , ; , since such a limit function exists; , and as ;
where is a nonnegative real sequence which converges to ; is subject to
for any given .Equations (46)-(47) also hold for by replacing everywhere appears. Consider
where .

Proof. Since is bounded in a right neighbourhood of for some real constant then the Jackson integral
converges to a function on which is a -antiderivative of and which is continuous at with and is a unique in this class of functions [39]. Note that ; ; since the series is convergent, since the Jackson integral (52) is convergent for each fix , and non-decreasing with since it is consists of nonnegative terms. It follows from (43), with the replacement , that
and hence (44) follows. By taking limits as in (44), one gets (46)–(48) since the limit exists for all by using Lemma 3(iv) for the expansion of the -derivative of order since as . Equation (49) follows by defining with for some ; for all i (≥N) and some , or equivalently
Hence, (46)–(48) follow from (55) for some strictly decreasing real sequence such that ; (≥N) for .

The quantum Caputo fractional solutions have explicit expressions as formulated in the subsequent result.

Theorem 11. Consider the left-sided -fractional Caputo solution of or fractional order of the functional differential system
with initial condition of the state being defined by -real vector functions , with and , and is a bounded piecewise continuous control function and .Thus, the unique left-sided solution of (55) is calculated almost everywhere via analytical expressions obtained from Lemma 10 as follows.(i)If so that then one gets for any given that
, where .(ii)If